SELECTED TOPICS ON BOOTSTRAP OR BACK TO STATISTICAL

SELECTED TOPICS ON BOOTSTRAP (OR: BACK TO STATISTICAL THEORY) DANA KANER M. SC. SEMINAR IN STATISTICS, JUNE 2017

APPROXIMATE LIKELIHOODS - INTRODUCTION

APPROXIMATE LIKELIHOODS - INTRODUCTION

THE EMPIRICAL LIKELIHOOD

THE EMPIRICAL LIKELIHOOD - EXAMPLE likelihood 1. 0 0. 8 0. 6 0. 4 0. 2 200 400 600 800 1000 1200

THE EMPIRICAL LIKELIHOOD - GOOD TO KNOW

APPROXIMATE PIVOT METHODS

APPROXIMATE PIVOT METHODS

APPROXIMATE PIVOT METHODS

APPROXIMATE PIVOT METHODS - EXAMPLE likelihood 1. 0 0. 8 0. 6 0. 4 0. 2 200 400 600 800 1000 1200

BOOTSTRAP PARTIAL LIKELIHOOD

BOOTSTRAP PARTIAL LIKELIHOOD

BOOTSTRAP PARTIAL LIKELIHOOD - EXAMPLE likelihood 1. 0 0. 8 0. 6 0. 4 0. 2 200 400 600 800 1000 1200

SUMMARY § Warning: It is important to note that in general, non of the described methods produce a true likelihood: a function that is proportional to the probability of a fixed event in the sample space. 1. 0 0. 8 likelihood 0. 6 0. 4 0. 2 200 400 600 800 1000 1200

PARAMETRIC AND NON-PARAMETRIC INFERENCE “Exact” Nonparametric Bootstrap Approximate Jackknife Infinitesimal Jackknife Nonparametric delta Parametric Bootstrap Fisher information Parametric delta

INFLUENCE FUNCTIONS

INFLUENCE FUNCTIONS

JK , IJ & NON PARAMETRIC BS

JK , IJ & NON PARAMETRIC BS “Exact” Nonparametric Bootstrap Parametric Bootstrap Approximate Jackknife Infinitesimal Jackknife Nonparametric delta Fisher information Parametric delta

THE PARAMETRIC BOOTSTRAP FOR MLE

THE PARAMETRIC BOOTSTRAP - EXAMPLE Normal model 10 0 50 0 20 40 60 80 100 Exponential model 300 200 100 0 20 40 60 80 100 140

PARAMETRIC MAX. LIKELIHOOD INFERENCE - THE BASICS

PARAMETRIC MAX. LIKELIHOOD INFERENCE - THE BASICS

PARAMETRIC MAX. LIKELIHOOD INFERENCE - THE BASICS

PARAMETRIC ML & BS INFERENCE, JACKKNIFE

PARAMETRIC ML, PARAMETRIC BS, IJ “Exact” Nonparametric Bootstrap Parametric Bootstrap Approximate Jackknife Infinitesimal Jackknife Nonparametric delta Fisher information Parametric delta

THE DELTA METHOD

THE DELTA METHOD

THE DELTA METHOD

THE DELTA METHOD FOR THE MEAN – EXAMPLE

THE DELTA METHOD’S CONNECTIONS “Exact” Nonparametric Bootstrap Approximate Jackknife Infinitesimal Jackknife Nonparametric delta Parametric Bootstrap Fisher information Parametric delta

THE EXPONENTIAL FAMILY

PROOF – THE PARAMETRIC CASE

PROOF – THE PARAMETRIC CASE

PROOF – THE PARAMETRIC CASE

Nonparametric Bootstrap Jackknife Infinitesimal Jackknife Nonparametric delta Parametric BS Fisher information Parametric delta